A Markov model for Selmer ranks in families of twists

Abstract

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve E over an arbitrary number field K. Under the assumption that Gal(K(E[2])/K) = S_3 we show that the density (counted in a non-standard way) of twists with Selmer rank r exists for all positive integers r, and is given via an equilibrium distribution, depending only on a single parameter (the `disparity'), of a certain Markov process that is itself independent of E and K. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.

This paper is a revised version of what was originally the second half of arXiv:1111.2321v1 [math.NT]

References in corpus (3)

• Random maximal isotropic subspaces and Selmer groups
• Disparity in Selmer ranks of quadratic twists of elliptic curves
• On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve

Cited by in corpus (7)

• Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field
• The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion
• Quadratic twists of abelian varieties and disparity in Selmer ranks
• Rank stability of elliptic curves in certain non-abelian extensions
• Rank growth of elliptic curves in $S_4$ and $A_4$ quartic extensions of the rationals
• On Conjectural Rank Parities of Quartic and Sextic Twists of Elliptic Curves
• Selmer stability for elliptic curves in Galois $\ell$-extensions